The Reward-Penalty-Selection Problem

TL;DR

This paper introduces the Reward-Penalty-Selection Problem, a new combinatorial optimization problem combining aspects of set cover and hitting set problems, with complexity analysis and specialized algorithms for certain cases.

Contribution

It defines the RPSP, analyzes its complexity, and provides polynomial and dynamic programming algorithms for specific instances, along with a graph theoretical generalization.

Findings

Complexity results for minimization and maximization variants.

Polynomial-time algorithm for laminar sets.

Dynamic programming solution for bounded tree-width instances.

Abstract

The Set Cover Problem (SCP) and the Hitting Set Problem (HSP) are well-studied optimization problems. In this paper we introduce the Reward-Penalty-Selection Problem (RPSP) which can be understood as a combination of the SCP and the HSP where the objectives of both problems are contrary to each other. Applications of the RPSP can be found in the context of combinatorial exchanges in order to solve the corresponding winner determination problem. We give complexity results for the minimization and the maximization problem as well as for several variants with additional restrictions. Further, we provide an algorithm that runs in polynomial time for the special case of laminar sets and a dynamic programming approach for the case where the instance can be represented by a tree or a graph with bounded tree-width. We further present a graph theoretical generalization of this problem and results regarding its complexity.